Complexity Theory Spring 2021


Charles E. Hughes
Computer Science
University of Central Florida


Structure: TR 0900-1015 (9:00AM-10:15AM); Virtual; 28 class periods, each 75 minutes long.

Class Zoom Link:

Go To Week
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14
, 15, 16

Instructor: Charles Hughes; Contact:; Use Subject COT6410
Office Hours: Tuesday 10:45AM-11:45AM; Thursday 12:15PM-1:30PM
OH Zoom Link:


GTA:  Daniel Daniel Gibney; Contact:; Use Subject COT6410
Office Hours: MW 1:30PM-3:00PM
OH Zoom Link:

Required Reading: All class notes linked from this site.

Recommended Reading

Web Pages:
Base URL:
Notes URL: Introductory Notes; Formal Language and Automata Theory; Computability Theory; Complexity Theory

Assignments: 6 for sure and maybe a seventh; Paper + Video Presentation with Slides

Exams: Midterm and Final.

Exam Dates (Tentative): Mid Term: Thursday, March 11; Withdraw Deadline:Friday, March 26; Spring Break: April 11-18; Final: Thurs., April 29, 7:00AM-9:50AM

Evaluation (Tentative):

  1. Mid Term: 125 points; Final Exam: 125 points (balance between weights will be adjusted in your favor)
  2. Assignments: 75 points;
  3. Individual Paper Reviews and Presentations: 125 points
  4. Extra -- 50 points used to increase weight of best exam,  always to your benefit
  5. Total Available: 500
  6. Grading will be  A >= 90%, B+ >= 85%, B >= 80%, C+ >= 75%, C >= 70%, D >= 50%, F < 50%; minus grades might be used.


Weeks#1: (1/12, 1/14) -- Syllabus; About Me; Preliminaries; Introduction; Formal Languages and Automata Theory
  1. Ground rules
  2. Decision problems
  3. Solving vs checking
  4. Procedures vs algorithms
  5. Introduction to theory of computation
  6. Terminology, goals and some historical perspective
  7. Overview of automata (finite, pushdown, linear bounded, Turing machines)
  8. Overview of formal languages (regular, context free, context sensitive, phrase structured)
  9. Regular Languages
  10. Closure Properties of Regular Languages
  11. Review material created by Prof. Jim Rogers, Earlam College
  12. Review material from Prof. David Workman, UCF Retired

Financial Aid (Assignment#1)

Survey at Webcourses
Due: Friday, January 15 at 11:59 PM

Week#2: (1/19, 1/21) -- Formal Languages and Automata Theory
  1. Continue Automata/Formal Languages Review
  2. State minimization using O(| Q | 2) table. Note alphabet is constant size.
  3. More closure properties
  4. Closure under min and max and discussions of various Reaching Algorithms (Depth-First Search)
  5. Pumping Lemma for Regular Languages (Pigeon Hole Principle)
  6. Myhill-Nerode Theorem
  7. Myhill-Nerode as proof of min DFA uniqueness
  8. Myhill-Nerode as a tool to show languages are not regular.
  9. The chosen language is
    L = { an bm | n is not equal to m}.
    This can be shown easily in an indirect manner by showing its complement is not regular. A direct approach is using right invariant equivalence classes as it is not amenable to the Pumping Lemma.
  10. Finite State Machines (Transducers): Mealy versus Moore Model
  11. Decision Problems for Regular Languages
  12. Introduction to Grammars (Regular and CFG mainly)
  13. Regular Grammars and Regular Languages


Week#3: (1/26, 1/28) -- Formal Languages and Automata Theory
  1. Context Free Grammars (A x where x (V ΣΣ)* )
  2. Use of context free grammars in parsing and notion of Ambiguity
  3. Review of reduced CFGs
  4. Chomsky Normal Form (CNF)
  5. The use of CNF in the Cocke-Kasami-Younger O(N3) parsing of CFLs generated by CNFs
  6. Pumping Lemma for CFLs
  7. Show L = { ww | w is in {a,b}+ } is not a CFL
  8. CSG for L
  9. Non-closure of CFLs under intersection and complement
  10. CFG for the complement of L
    { xy | |x| = |y| but x is not the same as y }
    can be first viewed as
    {x1 a x2 y1 b y2 | |x1|=|x2|, |y1|=|y2|} Union
      {x1 b x2 y1 a y2 | |x1|=|x2|, |y1|=|y2|}.
    But this can also be seen as
    {x1 a y1 x2 b y2 | |x1|=|x2|, |y1|=|y2|} Union
      {x1 b y1 x2 a y2 | |x1|=|x2|, |y1|=|y2|}.
    The above is easy to show as a CFL. We then union this with odd length strings and we have L1 complement.
  11. Closure of CFLs under substitution and intersection with Regular
  12. Decision problems for CFLs: is L(G) empty or finite/infinite are fine;
    Checking ambiguity, equality to Sigma*,  equivalence and non-empty intersection with another CFL are not
Assignment #2

See Webcourses (Assignment # 2) for description
Sample of Similar Problems with Solutions

Due: 2/9 (Key)

Week#4: (2/2, 2/4) -- Computability Theory
  1. Insights from intro to computability material
  2. Basic notions of computability and complexity
  3. Existence of unsolvable problems (counting and diagonalization)
  4. Solved, solvable (decidable, recursive), unsolved, unsolvable, re, non-re
  5. Hilbert's Tenth
  6. Undecidable problems made a bit more concrete
  7. Lots about problems and their complexity
  8. Halting Problem (HALT) is re, not decidable
  9. Set of algorithms (TOT) is non-re
  10. Halting Problem seen as fun
  11. Some consequences of non-re nature of algorithms
  12. Models of computation
  13. Systems related to FRS (Petri Nets, Vector Addition, Abelian Semi-Groups)
  14. RM simulated by Ordered FRS
  15. Primitive Recursive Functions (prf)
  16. Initial functions
  17. Closure under composition and recursion
  18. Primitive recursive functions SNAP, TERM, STP and VALUE
  19. Primitive Recursive Function in detail
  20. Addition and multiplication examples
  21. Sample functions and predicates
  22. Closure under cases
  23. Bounded minimization
  24. Arithmetic fuctions that use bounded search
  25. Pairing functions
  26. Limitations of primitive recursive
  27. mu-recursion and the partial recursive functions
  28. Notions of instantaneous descriptions
  29. Encodings
  30. Equivalence of models
  31. TMs to Register Machines
  32. RM to Factor Replacement Systems

Week#5: (2/9, 2/11) -- Computability Theory
  1. Factor Replacement to Recursive Functions
  2. Gory details on FRS to REC
  3. Universal machines
  4. Recursive Functions to TMs
  5. Consequences of equivalence
  6. Review Undecidability (Halting Problem, shown by diagonalization)
  7. RE sets and semidecidability
  8. The set of all re sets W0, W1, W2, ...
  9. Enumeration Theorem
  10. The set K = { n | n is in the n-th re set } = { n | n is in Wn } is re, non-recursive
  11. The set K0 = HALT = { <n,x> | x is in the n-th re set }
  12. Alternative characterizations of re sets
  13. Parameter Theorem (aka Sm,n Theorem)
    Assignment #3

    See Webcourses (Assignment # 3) for description
    Sample of Similar Problems with Solutions

    Due: 2/23 (Key)

Week#6: (2/16, 2/18) -- Computability Theory
  1. Quantification and re sets
  2. Quantification and Co-re sets
  3. Diagonalization revisited (set of algorithms is non-re and K0 (Lu) = HALT is non-rec)
  4. Reduction
  5. Classic sets Ko (Lu), NON-EMPTY (Lne), EMPTY (Le)
  6. Reduction from Ko that shows undecidability of K, HasZero, IsNonEmpty
  7. Complete re set (K and K0 as examples)
  8. Note: To be re-complete a set must be re
  9. Reduction from Ko that shows undecidability of K, HasIndentity, TOTAL, IsZero, IsEmpty, IsIdentity
  10. Equivalence of certain re sets (K, HasZero, IsNonEmpty, HasIndentity) to Ko
  11. Equivalence of certain non-re sets (IsZero, IsEmpty, IsIdentity) to TOTAL
  12. Quantification of Non-re, Non-Co-re sets
  13. Reducibility and degrees (many-one, one-one, Turing)
  14. Hierarchy or RE equivalence class (m-1 and 1-1 degrees)
  15. Rice's Theorem

        Assignment #4

    See Webcourses (Assignment # 4) for description
    Sample with key

        Due: 3/2 (Key)

Week#7: (2/23, 2/25) -- Computability Theory
  1. "Picture" proofs for Rice's Theorem
  2. Constant Time and Mortal Machines
  3. Introduction to rewriting systems (Thue, Post)
  4. Union, intersection, complement for recursive, re and non-re sets (can be, cannot be)
  5. Rewriting systems
  6. Post Canonical Forms
  7. Thue and Semi-Thue systems (relation to group theory)
  8. Word problems (Seni-Thue and Thue) and equivalence problems (Thue)
  9. Brief introduction to Post Correspondence Systems and Relation to Semi-Thue Systems


Week#8: (3/2, 3/4) -- Computability Theory
  1. Simulating Turing Machine by Semi-Thue System
  2. Simulating Turing Machines by Thue Systems
  3. Grammars and re sets
  4. Post Correspondence Problem (in detail)
  5. Unsolvable problems related to context-free grammars/languages
  6. Ambiguity of CFGs
  7. Non-Emptiness of CFL Intersections
  8. Context-Sensitive Grammars and Unsolvability Results
  9. Valid (CSL) and Invalid Traces (CFL)
  10. Intersection and Quotients of CFLs
  11. Details on Valid (CSL) and Invalid Traces (CFL)
  12. Intersection of CFLs revisited
  13. Quotients of CFLs revisited
  14. Type 0 grammars and Traces
  15. L =  Sigma*  for L a Regular or CFL
  16. L=L^2 for L a CFL
  17. Summary of Grammar Results
  18. Review session and sample exams
  19. Exam Topics
  20. Sample exam1; Sample exam1 key
  21. Sample exam2; Sample exam2 key
  22. Key to Samples from Notes
  23. Yet Other Examples (Focused on Formal Languages and Automata Theory)
  24. Yet Other Examples key (Focused on Formal Languages and Automata Theory)
  25. Midterm Legal Cheat Sheet

Week#9: (3/9, 3/11) -- Midterm Review and Exam

  1. Review (Tuesday)
  2. Midterm (Thursday)

Week#10 (3/16, 3/18: Complexity Theory

  1. Basics of Complexity Theory
  2. Decision vs Optimization Problems (achieving a goal vs achieving min cost)
  3. Polynomial == Easy; Exponential == Hard
  4. Polynomial reducibility
  5. Verifiers versus solvers
  6. P as solvable in deterministic polynomial time
  7. NP as solvable in non-deterministic polynomial time
  8. NP as verifiable in deterministic polynomial time
  9. Concepts of NP-Complete and NP-Hard
  10. Canonical NP-Complete problem: SAT (Satisfiability)
  11. Some NP problems that do not appear to be in P: SubsetSum, Hamiltonian Path, k-Clique
  12. Million dollar question: P = NP ?
  13. Construction that maps every problem solvable in non-deterministic polynomial time on TM to SAT
  14. SAT is polynomial reducible to (<=P) 3SAT
  15. 3SAT as a second NP-Complete problem
  16. Integer Linear Programming

Week#11: (3/23, 3/25, (3/26 is Withdrawal Deadline)) -- Complexity Theory
  1. Note: 3/26 is the Withdraw Deadline
  2. 3SAT <=P SubsetSum
  3. SubsetSum <=P Partition
  4. Partition equivalence to SubsetSum
  5. Discussion of group presentations
  6. Reduction of 3SAT to k-Vertex Cover
  7. 3-SAT to 3-Coloring
  8. Isomophism of k-Coloring with k-Register Allocation of live variables
  9. Scheduling problems introduced 
  10. Scheduling on multiprocessor systems
  11. Scheduling problems (fixed number of processors, minimize final finishing time)
  12. N processors, M tasks, no constraints
  13. Partition and scheduling problems
  14. Greedy heuristics
  15. 2-processor scheduling -- greedy based in list, sorted long to short, sorted short to long, optimal. Tradeoffs.
  16. Scheduling anomalies, level strategy for UET trees, level strategy for UET dags
  17. Precedences (lists, delays, preemption)
  18. Anomalies (reducing precedence, increasing processors, reducing times)
  19. Midterm Key
  20. You will each, individually, develop a paper and a presentation, based on an existing research paper, just as if you had to present to your peers and advisors. In addition to presenting the results and the way in which these were proven, you should comment on the paper's importance, readability, and replicability, and even its validity if you question that, just as if you were a journal or conference reviewer. Your report must be a tutorial on the topic of the paper so those with less time to delve into the paper can get a strong sense of the results and the context of those results. Specifying new open problems that you see as interesting is also a goal, but may not be attainable for some of these. The length of your paper is 6 to 12 pages double-spaced, 1" margins all around, using either Times Roman or Calibri 11 point, or Arial 10-point. The references are in addition to the narrative and appear on separate pages that do not count against the 6 to 12-page limit. Images may be used but if the total number of images exceeds a page, then all past that one-page aggregate will lead to a requirement for additional text. The presentation slides must be designed to support your 8 to 10-minute video. This means that there are likely 10 to 15 slides.
  21. Towards the end of the semester you give a presentation to a small group of other studesnt and they to you. This will be done in a breakout Zoom session where your presentation and any questions/answers are captured. You may, if you wish, create the presentation in advance and share a video, but you need to be there to participate in the Q&A and to hear other students' presentations. To do this, I assume you all have access to a webcam and use Zoom or a similar tool that captures your slides and you in one extended screen. Let me know if you lack a webcam. Note that you will be asked to provide some brief feedback to me on the papers of your fellow students taking part in teh same breakout.
  22. I will place about 50 sample papers out there at SampleTopics (not live yet). You may choose from any of these except for ones that have already been taken. You may also choose your own separate from these with permission from me. In general, the minimum page length in IEEE 2-column format is 8 pages or 10 pages in single column, single-spaced layout. The prefix CLAIMED_NAME means NAME has already chosen that. Please send me an email to assure you succeed in making your "claim."
  23. Realize that many of these suggested papers will be from arXiv and, while arXiv is a fantastic source, it is not refereed so if you question the validity of some result, that is actually a reasonable outcome so long as you present their approach and your counterarguments just as if you were a reviewer.
  24. This folder (to be made live later) contains some examples (papers and presentations) from past semesters.

              Final Paper and PowerPoint

                   Read above. Its weight is 125 points. The split is nominally 75 for the paper and 50 for your presentation
                   and feedback on other's presentations you provide.

              Due: 4/23

    Assignment #5

            See Webcourses (Assignment # 5) for description
            Sample with Key

  25.  Due: 4/6 (Key)

Week#12: (3/30, 4/1) -- Complexity Theory
  1. Unit Execution Time: Trees, forest, anti forests
  2. UET: DAGs and m=2
  3. Hamiltonian Path
  4. Traveling Salesman
  5. Knapsack (relation to SubsetSum), Dynamic Programming Approach
  6. Bin packing (fixed capacity, minimize number of bins)
  7. Pseudo polynomial-time solution for Knapsack using dynamic programming with changed parameters, n*W versus 2^n
  8. Tiling the plane and Bounded Tiling
  9. Bounded PCP
  10. Co-NP
  11. Reduction techniques
  12. P = co-P ; P contained in intersection of NP and co-NP
  13. NP-Hard
  14. QSAT as an example of NP-Hard, possibly not NP
  15. NP-Hard problems are in general functional not necessarily decision problems
  16. NP-Complete are decision problems as they are in NP
  17. NP-Easy -- these are problems that are polynomial when using an NP oracle
  18. NP-Equivalent is the class of NP-Easy and NP-Hard problems
  19. Optimization versions of SubsetSum and K-Color

            Assignment #6

                 See Webcourses (Assignment # 6) for description
                 Sample with Key

            Due: 4/20 (Key)

Week#13: (4/6, 4/8) -- Complexity Theory, More Notes
  3. ATM (Alternating NDTM)
  4. QSAT. Petri Nets, Presburger
  5. Validity of SubsetSum optimization reduction to SubsetSum Decision Problem Oracle
  6. 2SAT (linear time complexity)
  7. Uniform Min-1 is NP Equivalent
  8. Propositional Calculus
    Axiomatizable Fragments
    Unsolvability for Membership in Fragments of Diadic Partial Implicational Propositional Calculus
  9. Finite Convergence
    Finite Power Problem for CFLs

    Revisit Set Real-Time (Constant-Time)

    Real-Time and Mortal Machines

Week#14:  Spring Break


Week#15: (4/20, 4/22)

  1. FP is functional equivalent to P; R(x,y) in FP if can provide value y for input x via deterministic polynomial time algorithm
  2. FNP is functional equivalent to NP; R(x,y) in FNP if can verify any pair (x,y) via deterministic polynomial time algorithm
  3. TFNP is the subset of FNP where a solution always exists, i.e., there is a y for each x such that R(x,y).
  4. Factoring is in TFNP, but is it in FP?
  5. P = (NP intersect Co-NP) is interesting analogue to intersection of RE and co-RE but may not hold here
  6. It appears that TFNP does not have any complete problems!!!
  7. On 4/22 we will have the breakout sessions in which you give your presentations and hear those of a few other of your fellow students (typically five or six per breakout room)
  8. Final Exam Topics
  9. More stuff for final
  10. Sample Final Exam (Key)


Week#16:  (4/29 -- Final Exam)

  1. I will run  a help session on Tuesday, 4/27 at the normal class time
  2. All Project Papers and Presentations are Due on Tuesday, 4/23 by midnight
  3. Final Exam is Thursday April 29; 7:00AM to 9:50AM; There may also be a take home component


UCF (Charles E. Hughes)  -- Last Modified 1/26/2021